# Quaternion expansion

I have a quaternion equation $\psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$

Given conditions and data

1. Here P is a constant unit Quaternion defined for 3D rotation matrix as

$(p_1,p_2,p_3,p_4) , p_4\in R \tag 2$,$Y= p_1i+p_2j+p_3k,Y\in R^3 \tag 3$ ,Y is a vector

2. Vector $k(s)= \int_{0}^{s} \{((1-u)a_1+ub_1)i+((1-u)a_2+ub_2)j+((1-u)a_3+ub_3)k\}du\tag 4$

All $a_i,b_i$ are constants. I am bit confused with the exponential expansion of this vector. I got stucked because of that

Question

1. Basic definition of vector exponent is given as follows

$\begin{eqnarray} \exp( {\bf v}) & = & {\bf 1}+ {\bf v}+ \frac{1}{2!}{{\bf v}}^2 + \frac{1}{3!}{\bf v}^3 + \ldots \\ & = & [ {\bf 1}+ \frac{1}{2!}({\bf v}\cdot{\bf v}) + \frac{1}{4!}({\bf v}\cdot{\bf v})^2 + . . . ] + \\ & & \frac{{\bf v}}{\surd({\bf v}\cdot{\bf v})} [\surd({\bf v}\cdot{\bf v})+\frac{1}{3!}\surd({\bf v}\cdot{\bf v})^3+\ldots ] \\ & = & {\bf 1}\cosh(\surd({\bf v}\cdot{\bf v})) + \frac{{\bf v}}{\surd({\bf v}\cdot{\bf v})} \sinh(\surd({\bf v}\cdot{\bf v})), \\ & = & {\bf 1}\cosh(|{\bf v}|) + \frac{{\bf v}}{|{\bf v}|} \sinh(|{\bf v}|), \end{eqnarray}$

Is it possible to write $e^{\frac{1}{2}k(s)}$ using the above vector exponent explantion as a quaternion(beacuse of the form of final result ${\bf 1}\cosh(|{\bf v}|) + \frac{{\bf v}}{|{\bf v}|} \sinh(|{\bf v}|) )$? What could be the four elements of the quaternion $\psi(s)$? Shall we extract the four members of $\psi(s)$ from the product of matrix representation quaternion $P$ and quaternion $e^{\frac{1}{2}k(s)}$

• Your integral should be easy to compute. Then plug it into the formula you have for $\exp$. I don't see what the issue is. Also, we generally can't multiply or exponentiate vectors - call quaternions quaternions, not vectors. – whacka Aug 21 '14 at 9:45
• That was a type error.. I just made it correct just seconds before you replied. Please check again. Thanks – Nirvana Aug 21 '14 at 9:46
• "Your integral should be easy to compute. Then plug it into the formula you have for exp" That is ok.. But what is after that. beacuse I need to multiply that with P a quaterbion. But see the out put of exp. It is ${\bf 1}\cosh(|{\bf v}|) + \frac{{\bf v}}{|{\bf v}|} \sinh(|{\bf v}|) )$. How do I multiply that with P. By seeing that form $(t+vector)$. I thought I can write it as quaternion so that I can do multiplication easily with P using matrix representation of quaternion – Nirvana Aug 21 '14 at 9:50
• When you think of a vector $v$ as a quaternion, $v^2=-v\cdot v$, where the $\cdot$ is the usual dot product of $\Bbb{R}^3$. It looks like you are not conversant with quaternion operations. Study Wikipedia for starters. You see that because of the above sign error trig functions pop out instead of the hyperbolic ones. – Jyrki Lahtonen Aug 21 '14 at 10:07
• I am a starter in quaternions. I am bit familiar with basic operations . But here I am confused coz $Pe^{(\frac{1}{2}v)}$ Basically ,I am bit confused about how do I multiply P with $e^{(\frac{1}{2}v)}$. P is a quaternion but I am not sure what is $e^{(\frac{1}{2}v)}$. How to multiply a quaternion q with some thing like $e^{(\frac{1}{2}v)}$ – Nirvana Aug 21 '14 at 12:31

Just compute $k$ and use the exponential expansion of a quarternion. Then compute $\psi$ by multiplying out each component of it from the components of the exponential expansion with $P$.